International
Tables for Crystallography Volume A1 Symmetry relations between space groups Edited by Hans Wondratschek and Ulrich Müller © International Union of Crystallography 2006 |
International Tables for Crystallography (2006). Vol. A1. ch. 1.5, p. 38
Section 1.5.6.1. General results
a
Abteilung Reine Mathematik, Universität Ulm, D-89069 Ulm, Germany |
The first very easy but useful remark applies to general groups :
Remark . Let be a maximal subgroup of of finite index . Then . Hence the maximality of implies that either and is a normal subgroup of or and has i maximal subgroups that are conjugate to .
The smallest possible index of a proper subgroup is 2. It is well known and easy to see that subgroups of index 2 are normal subgroups:
Proof . Choose an element , . Then . Hence and therefore . Since this is also true if , the proposition follows. QED
Let be a subgroup of a group of index 2. Then is a normal subgroup and the factor group is a group of order 2. Since groups of order 2 are Abelian, it follows that the derived subgroup of (cf. Definition 1.5.5.2.1) (which is the smallest normal subgroup of such that the factor group is Abelian) is contained in . Hence all maximal subgroups of index 2 in contain . If one defines , then is an elementary Abelian 2-group and hence a vector space over the field with two elements. The maximal subgroups of are the maximal subspaces of this vector space, hence their number is , where .
This shows the following:
Dealing with subgroups of index 3, one has the following:
Proposition 1.5.6.1.3. Let be a subgroup of the group with . Then is either a normal subgroup of or and there are three subgroups of conjugate to .
Proof . is isomorphic to a subgroup of that acts primitively on . Hence either and is a normal subgroup of or , and there are three subgroups of conjugate to . QED